Paraconsistent Quasi-Set Theory

Abstract

Paraconsistent logics are logics that can be used to base inconsistentbut non-trivial systems. In paraconsistent set theories, we can quan-tify over sets that in standard set theories (that are based on classicallogic), if consistent, would lead to contradictions, such as the Russell set,R = fx : x =2 xg. Quasi-set theories are mathematical systems built fordealing with collections of indiscernible elements. The basic motivationfor the development of quasi-set theories came from quantum physics,where indiscernible entities need to be considered (in most interpreta-tions). Usually, the way of dealing with indiscernible objects within clas-sical logic and mathematics is by restricting them to certain structures,in a way so that the relations and functions of the structure are not sufficient to individuate the objects; in other words, such structures are notrigid. In quantum physics, this idea appears when symmetry conditionsare introduced, say by choosing symmetric and anti-symmetric functions(or vectors) in the relevant Hilbert spaces. But in standard mathematics,such as that built in Zermelo-Fraenkel set theory (ZF), any structure canbe extended to a rigid structure. That means that, although we can dealwith certain objects as they were indiscernible, we realize that from out-side of these structures these objects are no more indiscernible, for theycan be individualized in the extended rigid structures: ZF is a theoryof individuals, distinguishable objects. In quasi-set theory, it seems thatthere are structures that cannot be extended to rigid ones, so it seems thatthey provide a natural mathematical framework for expressing quantumfacts without certain symmetry suppositions. There may be situations,however, in which we may need to deal with inconsistent bits of infor-mation in a quantum context, even if these informations are concernedwith ways of speech. Furthermore, some authors think that superposi-tions may be understood in terms of paraconsistent logics, and even thenotion of complementarity was already treated by such a means. This is,apparently, a nice motivation to try to merge these two frameworks. Inthis work, we develop the technical details, by basing our quasi-set theoryin the paraconsistent system C1. We also elaborate a new hierarchy ofparaconsistent calculi, the paraconsistent calculi with indiscernibility. Forthe finalities of this work, some philosophical questions are outlined, butthis topic is left to a future work.